When strong winds rapidly carry air down from mountains to a lower elevation, the air has no time to exchange heat with its surroundings. The air is compressed as the pressure rises, and its temperature can increase dramatically. These warm winds are called Chinook winds in the Rocky Mountains and Santa Ana winds in California. Suppose the air temperature high in the mountains behind Los Angeles is $0°\mathrm{C}$ at an elevation where the air pressure is $60\mathrm{kPa}$. What will the air temperature be, in ${}^{\circ }\mathrm{C}$ and ${}^{\circ }F$ when the Santa Ana winds have carried this air down to an elevation near sea level where the air pressure is $100\mathrm{kPa}$?

The air temperature high in the mountains behind Los Angeles is $0°\mathrm{C}$

At an elevation where the air pressure is role="math" localid="1648112467675" $=60kPa.$

Air pressure $=100kPa$

From a rearrangement of the ideal gas law, we have that

$\frac{p_1{V}_1}{T_1}=\frac{p_2{V}_2}{T_2}$

This means that we can express the temperature after the process in terms of the initial temperature as,

$T_2=\frac{p_2{V}_2}{p_1{V}_1}{T}_1$

Since the air has no time to exchange heat, the process is adiabatic. This means that the pressure times the volume to the power of $y$remains constant. That is,

$p_1{V}_1^{\gamma }=p_2{V}_2^{\gamma }\Rightarrow \frac{p_1}{p_2}={\left(\frac{V_2}{V_1}\right)}^{\gamma }$

By the power rules $(\sqrt[a]{b}=b^{\frac{1}{a}}$and $\left.b^a=(1/b{)}^{-a}\right)$, we can write that

$\frac{V_2}{V_1}={\left(\frac{p_2}{p_1}\right)}^{-\frac{1}{\gamma }}$

Combining these, we can express the temperature after the adiabatic compression as

$T_2=\frac{p_2}{p_1}{\left(\frac{p_2}{p_1}\right)}^{-\frac{1}{\gamma }}{T}_1={\left(\frac{p_2}{p_1}\right)}^{\gamma -\frac{1}{\gamma }}{T}_1={\left(\frac{p_2}{p_1}\right)}^{\frac{\gamma -1}{\gamma }}{T}_1$

We can now substitute our numerical values and find

$T_2={\left(\frac{100}{60}\right)}^{\frac{1.40-1}{1.40}}·273=316\mathrm{K}$

In Celsius, this temperature will be $43°\mathrm{C}$In Fahrenheit , this temperature will be$109°\mathrm{F}$.